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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.inv_hyper_over"></a><a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">Inverse Hyperbolic
      Functions Overview</a>
</h3></div></div></div>
<p>
        The exponential function is defined, for all objects for which this makes
        sense, as the power series
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb1.svg"></span>

        </p></blockquote></div>
<p>
        with <span class="emphasis"><em><code class="literal">n! = 1x2x3x4x5...xn</code></em></span> (and <span class="emphasis"><em><code class="literal">0!
        = 1</code></em></span> by definition) being the factorial of <span class="emphasis"><em><code class="literal">n</code></em></span>.
        In particular, the exponential function is well defined for real numbers,
        complex number, quaternions, octonions, and matrices of complex numbers,
        among others.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="emphasis"><em><span class="bold"><strong>Graph of exp on R</strong></span></em></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/exp_on_r.png"></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="emphasis"><em><span class="bold"><strong>Real and Imaginary parts of exp on C</strong></span></em></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/im_exp_on_c.png"></span>
        </p></blockquote></div>
<p>
        The hyperbolic functions are defined as power series which can be computed
        (for reals, complex, quaternions and octonions) as:
      </p>
<p>
        Hyperbolic cosine:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb5.svg"></span>

        </p></blockquote></div>
<p>
        Hyperbolic sine:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb6.svg"></span>

        </p></blockquote></div>
<p>
        Hyperbolic tangent:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb7.svg"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="emphasis"><em><span class="bold"><strong>Trigonometric functions on R (cos: purple;
          sin: red; tan: blue)</strong></span></em></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/trigonometric.png"></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="emphasis"><em><span class="bold"><strong>Hyperbolic functions on r (cosh: purple;
          sinh: red; tanh: blue)</strong></span></em></span>
        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/hyperbolic.png"></span>
        </p></blockquote></div>
<p>
        The hyperbolic sine is one to one on the set of real numbers, with range
        the full set of reals, while the hyperbolic tangent is also one to one on
        the set of real numbers but with range <code class="literal">[0;+∞[</code>, and therefore
        both have inverses.
      </p>
<p>
        The hyperbolic cosine is one to one from <code class="literal">]-∞;+1[</code> onto
        <code class="literal">]-∞;-1[</code> (and from <code class="literal">]+1;+∞[</code> onto <code class="literal">]-∞;-1[</code>).
      </p>
<p>
        The inverse function we use here is defined on <code class="literal">]-∞;-1[</code>
        with range <code class="literal">]-∞;+1[</code>.
      </p>
<p>
        The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent,
        and can be computed as
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb15.svg"></span>

        </p></blockquote></div>
<p>
        The inverse of the hyperbolic sine is called the Argument hyperbolic sine,
        and can be computed (for <code class="literal">[-1;-1+ε[</code>) as
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb17.svg"></span>

        </p></blockquote></div>
<p>
        The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine,
        and can be computed as
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb18.svg"></span>

        </p></blockquote></div>
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